Vacuum expectation value

The vacuum expectation value of an odd number of (Heisenberg) photon operators vanishes. 

Spectral Representation.—As an application of the invariance properties of quantum electrodynamics we shall now use the results obtained in the last section to deduce a representation of the vacuum expectation value of a product of two fermion operators and of two boson operators. The invariance of the theory under time inversion and more particularly the fact that... 

Similar results can be obtained for the vacuum expectation value of electromagnetic field operators. The spectral representation takes the form... 

Consider next the current operator ju(x). The correspondence principle suggests that its form is j ( ) = — (e/2)( ( )yB, (a )]. Such a form foijn(x) does not satisfy Eq. (11-477). In fact, due to covariance, the spectral representation of the vacuum expectation value of 8u(x)Av(x), where ( ) is an arbitrary four-vector, is given by... 

The second Higgs field acts in such a way that if the vacuum expectation value is zero, ( ) = 0, then the symmetry breaking mechanism effectively collapses to the Higgs mechanism of the standard SU(2) x U(l) electroweak theory. The result is a vector electromagnetic gauge theory 0(3)/> and a broken chiral SU(2) weak interaction theory. The mass of the vector boson sector is in the A(3) boson plus the W and Z° particles. 

The element (p apaq asar]v) may be written as a vacuum expectation value of a string of creation and annihilation operators such as contracted terms, each of which depends only on the overlap between the orbitals. Since these are 5pq for orthonormal orbitals, we see that the vacuum... 

Due to the fact that the Lagrangian incorporates the creation and destruction of field quanta, not even the time-development of a single particle is a simple matter. The time development can be expressed in terms of the electron (fermion) and photon propagators, which are defined as the vacuum expectation values of the time-ordered product of field operators. For the fermions one has... 

Liouvillean space the expectation values become simple matrix elements, no longer a trace, and may even be viewed as thermal vacuum expectation values with respect to the given initial thermodynamic state. We can now restate Eq. (32) as a stationary action principle in this superspace ... 

Since the left- and right-hand states may be written simply as single annihilation and creation operators acting on the vacuum, the desired matrix element of A may be rewritten as the vacuum expectation value of a new operator, = a Aal. Therefore, we need only rewrite B in normal order and select only the terms that contain no annihilation or creation operators, as we did in Eq. [77]. After much algebraic manipulation, which we shall omit here, it can be shown that... 

Fiow does Wick s theorem help us in evaluating matrix elements of second-quantized operators Recall that any matrix element of an operator may be written as a vacuum expectation value by simply writing its left- and right-hand determinants as operator strings acting on the vacuum state, I ). The... 

Note that the first and second terms on the right-hand side of this equation are simply the spin-orbital Fock operator (in normal-ordered form), and the last two terms are the Hartree-Fock energy (i.e., the Fermi vacuum expectation value of the Hamiltonian). Thus, we may write... 

The Feynman propagator accounts for the motion of an electron or positron from a space-time point Xj = (ctj, xj) to another point 2 = ct2, X2). It is defined by the vacuum expectation value... 

Due to the redefinition of the vacuum (<4 0s) = 0) the Hamiltonian is still not bounded from below. This property must be implemented by a renormalization of the energy scale, i.e. by subtraction of the vacuum expectation value of Ha,... 

In this case, however, the vacuum expectation value does not vanish. 

The operator (175) measures the energy of a given state with respect to the vacuum IO5) in the presence of the external potential. In the noninteracting situation these energy differences correspond directly to the observable ionization potentials. However, the operator (175) does not yet reflect the fact that the vacuum energies resulting from different external potentials are not identical (Casimir effect). The differences between the vacua are most easily seen on a local scale The vacuum expectation value of the current density operator (7) reads... 

Due to this form invariance of the Lagrangian under the renormalization prescription the stracture of the theory, which e.g. expresses itself in Dyson s equations and Ward-Takahashi identities, remains completely unchanged. The renormalized Greens functions, i.e. the vacuum expectation values of f and A, are thus simply obtained from... 

Substituting (25) into (23) we are led to evaluate the vacuum-vacuum expectation values of products of creation and destruction operators contained in V t) or introduced by the trace. We use the well-known theorem of Wick, which expresses such an element as the sum of all possible products of "contracted pairs of operators. This rule becomes identical with the one given in Section II-A for the evaluation of , if the statistical averages < 6> are replaced by the vacuum averages or "contractions <0 a6 0> ... 

To illustrate the use of the Goldstone program, let us consider the wave operator and correlation energies of some atom or molecule. For close-shell systems, of course, we can choose always a 1-dimensional model space, 0c), which coincides with the reference state total energy of the system up to first order is equal to the vacuum expectation value of the normal-ordered Hamiltonian... 

For all higher-order correlation energies (of closed-shell systems), moreover, we can follow similar lines and evaluate the vacuum expectation values EI"I = (0c HX2 " 0c). This results in the standard Moeller-Plesset expressions for the energy corrections. 

In the Standard Model (SM), there is another fundamental parameter with the dimension of mass—the Higgs vacuum expectation value (VEV), which determines the electroweak unification scale. The electron mass mg and quark masses mq are proportional to the Higgs VEV. Consequently, the dimensionless parameters Xe = wte/AqcD and Xq = mq/Agco link the electroweak unification scale with the strong scale. For the light u and d quarks, Xq -c 1. As a result, the proton mass wip is proportional to Aqcd and hence X is proportional to p. In what follows, we use p instead of Xg as it is more directly linked to the experimentally accessible atomic and molecular observables. 

The last necessary ingredient toward a well-defined and consistent quantum field theory of radiation interacting with a fermionic matter field is the introduction of normal-ordered products of field operators. The necessity for this step is easily realized by consideration of the vacuum expectation value of Gauss law. 

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